Econ 600 Schroeter Fall 2005 Exam #1 Do all three problems. Weights: #1 - 25%, #2 - 35%, #3 - 40%. Closed book, closed notes. Be sure your answers are presented in a neat and well-organized manner. 1. Answer True or False for each of the following statements. If the statement is false, indicate how it could be changed to a true statement with a small change in the wording. a. is concave if and only if sets of the form ℜ→ℜnF :(){}axFxn≥ℜ∈ : are convex for all . ℜ∈a b. Given differentiable. ℜ→ℜnF :()⋅F is quasi-concave if and only if ()( )0≥−∂∂vuvxF for all such that nvu ℜ∈,()(.vFuF ≥) c. Consider the problem: 2...max xxtrw such that [].1,1−∈x 1*1−=x and are both strict local solutions but neither is a strict global solution. 1*2=x d. Let be differentiable and consider the problem: ℜ→ℜnF : ().min...xFxtrw If ()0*=∂∂xxF and (*22x)xF∂∂ is positive semi-definite, then is a strict local solution. *x e. If is strictly concave then ℜ→ℜnF :()⋅F is strictly quasi-concave.22. Let U be a convex subset of Prove that the function .nℜℜ→UG : is convex if and only if ()(){}xGyandUxyxn≥∈ℜ∈+:,1 is a convex set in . 1+ℜn 3. Let be differentiable with negative definite Hessian throughout . Consider the problem: ℜ→ℜ2:F2ℜ ()()()*,,,max2121,...21bxxgthatsuchxxFxxtrw= where b is a scalar constant and ()⋅g is linear so that ()221121, xgxgxxg+= for scalar constants g1 and g2, not both zero. a. Write down the Lagrangian and the first order necessary conditions (FONC) for to be a local solution to problem (*). 2*ℜ∈x b. Show that any that satisfies the FONC is a strict local solution to problem (*). Do this by showing that the second order sufficient (bordered Hessian) condition necessarily holds at , given the assumptions that we have made about *x*x()⋅F and . ()⋅g (Note: Actually, an that satisfies the FONC is, in this case, a strict global solution to problem (*), but you don't have to show that.)